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Chapter 2: Problem 124

Explain how to find the difference quotient of a function \(f\) \(\frac{f(x+h)-f(x)}{h},\) if an equation for \(f\) is given.

Short Answer

Expert verified
The difference quotient of function \(f(x)\), is calculated by the formula \(\frac{f(x+h)-f(x)}{h}\). First, replace \(x\) in the function \(f(x)\) with \(x+h\) and \(x\). Then, substitute the expressions obtained, into the difference quotient formula. Simplify the resulting fraction by canceling the common factors, such as \(h\), to obtain the final difference quotient.

Step by step solution

01

Understand the Difference Quotient

The difference quotient of a function \( f \) is the expression \(\frac{f(x+h) - f(x)}{h}\). Here, \( h \) is a small increment in \( x \), and \( f(x+h) \) and \( f(x) \) represent the values of the function at points \( x+h \) and \( x \) respectively.
02

Substitute \(x+h\) and \(x\) in Function \(f(x)\)

To determine the difference quotient, replace \(x\) in the function \(f(x)\) with \(x+h\) and \(x\). That will give you \(f(x+h)\) and \(f(x)\).
03

Insert \(f(x+h)\) and \(f(x)\) in the Difference Quotient

Now substitute the expressions of \(f(x+h)\) and \(f(x)\) obtained in Step 2, in the difference quotient formula. This will result in a fraction.
04

Simplify the Expression

The expression you get in Step 3 can probably be simplified. Try to simplify the numerator ensuring that it can be factored by \(h\). Once that's done, you can cancel the \(h\) in the numerator with the \(h\) in the denominator.
05

Check Your Answer

And that is the difference quotient of function \(f(x)\). But always remember to check your work.

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